Discriminant feature extraction is an important topic in pattern recognition and classification. Current approaches used for linear discriminant feature extraction include Principal Component Analysis (PCA) and Linear Discriminant Analysis (LDA). Applications for PCA and LDA include pattern recognition and computer vision. These methods use a vector-based representation and compute scatters in a Euclidean metric, i.e., an assumption is made that the sample space is Euclidean, where an example metric is a function that computes a distance or similarities between two points in a sample space.
Despite the utility of these subspace learning algorithms, the reliance on a Euclidean assumption of a data space when computing a distance between samples has drawbacks, including the potential of a singularity in a within-class scatter matrix, limited available projection directions, and a high computational cost. Additionally, these subspace learning algorithms are vector-based and arrange input data in a vector form regardless of an inherent correlation among different dimensions in the data.
In one nonlinear approach, Linear Laplacian Discrimination (LLD), weights are introduced to scatter matrices to overcome the Euclidean assumption, however, the weights are defined as a function of distance and therefore still use an a priori assumption on a metric of the sample space.